Scaling relations for percolation in the 2D high temperature Ising Model (Applications of the Renormalization Group Methods in Mathematical Sciences)

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Scaling relations for percolation in the $2D$ high temperature Ising Model

神戸大学大学院理学研究科樋口保成 (Yasunari Higuchi)

Graduate school of Science,

Kobe University

大阪電気通信大学工学部竹居正登 (Masato Takei)

Faculty of Engineering, Osaka Electro-Communication University

コロラド大学 Yu Zhang

University of Colorado

1 Ising model

1.1 Definition

We define the Ising model on the two-dimensional square lattice $\mathbb{Z}^{2}$:

The spin

con-figuration space is denoted by $\Omega$ $:=\{-1, +1\}^{\mathbb{Z}^{2}}$ This model has two parameters; the

temperature $T\in[0, \infty)$ and the external magnetic field $h\in \mathbb{R}$

.

For a finite region

$V\subset \mathbb{Z}^{2}$ and a

boundary condition $\omega\in\Omega$, the interaction energy for aspinconfiguration $\sigma\in\Omega_{V}$ $:=\{-1, +1\}^{V}$ is given by

$H_{V,h}^{\omega}( \sigma):=-\frac{1}{2}\sum_{u,v\in V,|u-v|=1}\sigma(u)\sigma(v)-\sum_{v\in V}(h+\sum_{u\not\in V,|u-v|=1}\omega(u))\sigma(v)$,

where $|x|$ $:=|x^{1}|+|x^{2}|$ for$x=(x^{1}, x^{2})\in \mathbb{Z}^{2}$. The function $H_{V,h}^{\omega}(\sigma)$ iscalled Hamiltonian.

The

_finite

volume Gibbs distribution is defined by

$q_{V,T,h}^{\omega}(\sigma)=(Z_{V,T,h}^{\omega})^{-1}\exp\{-H_{V,h}^{\omega}(\sigma)/(\mathfrak{K}T)\},$

where

$Z_{V,T,h}^{\omega}:= \sum_{\sigma’\in\Omega_{V}}\exp\{-H_{V,h}^{\omega}(\sigma’)/(\mathfrak{K}T)\}$

is a normalizing constant called the partition function, and $\mathfrak{K}$ denotes the Boltzmann

constant.

Let $\mathcal{F}_{V}$denotethe a-algebra generated bythe spin variables in$V\subset \mathbb{Z}^{2}$, and

$\mathcal{F}:=\mathcal{F}_{\mathbb{Z}^{2}}.$

Then

we

have

$q_{V,T,h}^{\omega}[\sigma(v)=+1|\mathcal{F}_{\{v\}^{c}}](\sigma)$

$=[1+ \exp\{\frac{-2}{\mathfrak{K}T}(h+\sum_{u\in V,|u-v|=1}\sigma(u)+\sum_{u\not\in V,|u-v|=1}\omega(u))\}]^{-1}$

Note that the

case

$T=\infty$ corresponds to the _{independent site} _percolation _problem:

Letting $Tarrow\infty$ with $h/(\mathfrak{K}T)arrow H$, we can see that $\sigma(v)=+1$ with probability _$p=$

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1.2 Phase

transition

in

Ising

model

A probability

measure

$\mu$

on

$\Omega$ is called the Gibbs

measure

if it satisfies the following

Dobrushin-Lanford-Ruelle

equation:

$\mu(\cdot|\mathcal{F}_{V^{c}})(\omega)=q_{V,T,h}^{\omega}(\cdot)$ for $\mu$-almost all $\omega.$

$\bullet$ Every limit point of the finite volume Gibbs distribution $q_{V,T,h}^{\omega}$

as

$V\nearrow \mathbb{Z}^{2}$ is a

Gibbs

measure.

$\bullet$ By using stochastic monotonicity, the following limiting Gibbs

measure

with pure

boundary conditions exist:

$\mu+:=\lim_{V\nearrow \mathbb{Z}^{2}}q_{V,T,h}^{+}$, and $\mu-:=\lim_{V\nearrow \mathbb{Z}^{2}}q_{\overline{V},T,h}.$

Both $\mu+$ and $\mu_{-}$ are invariant under spatial translations. Moreover, we have

$\mu_{-}\leq\mu\leq\mu+$ for any Gibbs

measure

$\mu.$

$\bullet$ The set of Gibbs

measures

is a convex set, and its extremal points correspond to

‘pure phases’; $\mu+$ and $\mu$-are among them.

$\bullet$ There exists the critical temperature $T_{c}\in(0, \infty)$:

$r$ If $T<T_{c}$ and $h=0$, then $\mu+\neq\mu_{-}.$

$\nu$ Otherwise not only _{$\mu+=\mu_{-}$} but also there is a unique Gibbs

measure.

Aizenman (1980), and Higuchi (1981) showed that for the Ising model on $\mathbb{Z}^{2}$, there

are

only two extremal Gibbs

measures

$\mu+$ and $\mu_{-}$, and for any Gibbs

measure

$\mu$, there

is

an

$\alpha\in[0,1]$ such that

$\alpha\mu_{+}+(1-\alpha)\mu_{-}.$

For higher dimensional cases, there

are

non translation-invariant Gibbs

measures

(Do-brushin (1972)$)$ , but every translation-invariant Gibbs

measure

isa

convex

combination

of $\mu+$ and $\mu_{-}$ (Bodineau (2006)).

2 Percolation

in

the high-temperature Ising model

We consider the percolation problem in the high-temperature regime. (See Higuchi (1997) for a survey.) First

we

prepare basic terminologies for the percolation theory.

$\bullet$ $A$ path [resp. $(*)$-path] is a sequence $x_{1},$ $x_{2},$ _$\ldots,$$x_{s}$ in

$\mathbb{Z}^{2}$ with

$|x_{i}-x_{i-1}|=1$ [resp.

$|x_{i}-x_{i-1}|_{\infty}=1]$ for $1<i\leq s$, where $|x|_{\infty}$ $:= \max\{|x^{1}|, |x^{2}|\}$ for $x=(x^{1}, x^{2})\in \mathbb{Z}^{2}.$

$\bullet$ $A$ path

on

which all spin variable

are

$+$ is called $a(+)$-path. We define $(-*)$-path

in a sirnilar

manner.

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$\triangleright$ More generally, for _$V,$$V’\subset \mathbb{Z}^{2}$, let $\{Vrightarrow+V’\}$ denote the event that some

point in $V$ is connected by _$a(+)$-path to some point in $V’.$ $\bullet$ $A$ sequence

$x_{1},$ $x_{2},$_$\ldots,$$x_{s}$ in

$\mathbb{Z}^{2}$ is called

a circuit if

$\{(i, j):|x_{i}-x_{j}|=1\}=\{(i, j):|i-j|=1 or \{i,j\}=\{1, s\}\}.$

We define $a(+)$-circuit and $a(-*)arrow$circuit

as

above.

$\bullet$ The $(+)$-cluster containing $x\in \mathbb{Z}^{2}$ is defined by $C_{x}^{+}:=\{y\in \mathbb{Z}^{2} : xrightarrow+y\}$. We

often write $\{xrightarrow+\infty\}$ for _{$\{\#C_{x}^{+}=\infty\}$}

.

We adopt a similar notation for –spins. $\bullet$ For each extremal Gibbs measure

$\mu$, it is known that $\mu(\bigcup_{x\in \mathbb{Z}^{2}}\{\#C_{x}^{+}=\infty\})$ is

either $0$ or 1. If it is equal to 1, then we say that _$(+)$-percolation occurs. We define

$(-)$-percolation similarly.

When $T>T_{c}$, there exists a unique Gibbs measure for each $h\in \mathbb{R}$:

For every$\omega\in\Omega,$

$\lim_{V\nearrow Z^{2}}q_{V,T,h}^{\omega}=\mu_{T,h}.$

The origin in $\mathbb{Z}^{2}$

is denoted by O. We write $C_{0}^{+}$ for the _$(+)$-cluster containing $O$, and define

$h_{c}(T) := \inf\{h : \mu_{T,h}(\#C_{0}^{+}=\infty)>0\}.$

It is shown in Higuchi (1993a) that $h_{c}(T)>0$ whenever $T>T_{c}$. (It is also known that

$h_{C}(T)=0$ for $T\leq T_{c}.$) When $T>T_{c}$, the percolation transition at $h=h_{c}(T)$ is sharp

(Higuchi $(1993b)$). Hereafter we fix a $T>T_{c}$, and abbreviate $\mu_{T,h}$ to $\mu_{h}$, and $h_{c}(T)$ to

$h_{c}$

.

The expectation under

$\mu_{h}$ is denoted by $E_{h}.$

3 Scaling relations

We investigate the critical behavior of principal quantities in percolation. We adopt the following notation:

$\bullet$ $f(n)\approx n^{\zeta}$

means

that _{$\lim_{narrow\infty}\frac{\log f(n)}{\log n}=\zeta.$}

$\bullet$ $f(n)_{\wedge}^{\vee}g(n)$

means

that $C_{1}g(n)\leq f(n)\leq C_{2}g(n)$ for some positive constants _{$C_{1}$}

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3.1 Conjectured power laws and scaling relations

[Near the critical point]

$\blacksquare$ The percolation probability $\theta(h);=\mu_{h}(\#C_{0}^{+}=\infty)$.

$\theta(h)\approx(h-h_{c})^{\beta}$

as

$h\searrow h_{c}.$

$\blacksquare$ The expected size of the finite cluster $\chi(h)$ $:=E_{h}[\#C_{0}^{+}:\#C_{0}^{+}<\infty].$ $\chi(h)\approx|h$一 as $h\searrow h_{c}.$

$*$ Gap exponent $\triangle$

For any $k\geq 2$,

as

$harrow h_{c}.$

$\blacksquare$ Correlation length

$\xi(h):=[\frac{1}{\chi(h)}\sum_{v\in Z^{2}}|v|^{2}\mu_{h}(Orightarrow+v, \#C_{0}^{+}<\infty)]^{1/2}$

$(h)\approx|h$一。 as $harrow h_{c}.$

[At the critical point] Let $S(n):=[-n, n]^{2}$ _and $\partial S(n):=\{x\in \mathbb{Z}^{2}:|x|_{\infty}=n\}.$

$\blacksquare$ The 1-arm probability _{$\pi_{h}(n)$} $:=\mu_{h}(Orightarrow+\partial S(n))$. $\pi_{h_{c}}(n)\approx n^{-1/\delta_{r}}$ as $narrow\infty.$

$\blacksquare$ The connectivity function $\tau_{h}(n);=\mu_{h}(Orightarrow+(n, 0))$

.

$\blacksquare$ The size distribution of the finite cluster

$\mu_{h_{c}}(\#C_{0}^{+}\geq n)\approx n^{-1/\delta}$

ae

$narrow\infty.$

[Scaling relations]

$\beta(\delta+1)=\gamma+2\beta=\triangle+\beta, \gamma=\nu(2-\eta)$

.

[Hyperscaling relations] (only for small d)

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3.2 Finite-size

scaling correlation

length

Let $A^{+}(n, m)$ be the event that there exists$a(+)$-path between theleft side and right

side of $[-n, n]\cross[-m, m]$_; _the $(+)$-path is called the horizontal $(+)$-crossing. Similarly

we define $A^{-*}(n, m)$

.

The

finite-size

scaling correlation length$L(h, \epsilon_{0})$ by

$L(h, \epsilon_{0}):=\{\begin{array}{ll}\min\{n: \mu_{h}(A^{+}(n, n))\geq 1-\epsilon_{0}\} (h>h_{c}) ,\infty (h=h_{c}) ,\min\{n :\mu_{h}(A^{+}(n, n))\leq\epsilon_{0}\} (h<h_{c}) .\end{array}$

Here $\epsilon_{0}$ is

a

small positive constant. It is known that $\xi(h)_{\wedge}\vee L(h, \epsilon 0)$

.

The following theorems play important roles in deriving scaling relations. For the independent percolation, these theorems are proved by Kesten (1987b).

Theorem 3.1. (i) For any $n<L(h, \epsilon 0),$ $C_{1} \leq\frac{\pi_{h}(n)}{\pi_{h_{。}}(n)}\leq C_{2}.$

(ii) As $h\searrow h_{c},$

$\theta(h)_{\wedge}\cdot\pi_{h}(L(h, \epsilon_{0}))_{\wedge}^{\vee}\pi_{h_{c}}(L(h, \epsilon_{0}))$.

From this theorem, we can obtain a scaling relation involving $\beta,$$\delta_{r}$, and

$v$, if they

exist.

Theorem 3.2. (i) For $t>1$,

as

$harrow h_{c},$

$E_{h}[(\#C_{0}^{+})^{t}:\#C_{0}^{+}<\infty]\wedge L(h, \epsilon_{0})^{2t}\pi_{h}$

。

$(L(h, \epsilon 0))^{t+1}.$

(ii) For $t=1$, as $harrow h_{c},$

$E_{h}[\#C_{0}^{+}:\#C_{0}^{+}<\infty]\approx L(h, \epsilon 0)^{2}\pi_{h}$

。

$(L(h, \epsilon 0))^{2}.$

This theorem suggests thatthe volumeofa “large critical cluster” in $S(n)\approx n^{2}\pi_{h_{c}}(n)$

.

Theorem 3.3. As $x\searrow 0,$ $L(h_{c}-x, \epsilon_{0})=L(h_{C}+x, \epsilon_{0})$

.

This implies the symmetry of critical exponents on the left and right of $h_{c}.$

3.3

Scaling

relations

We can show the following relations between critical exponents (Higuchi, Takei, and Zhang (2010, 2011)$)$; those

are

obtained by Kesten $(1986, 1987a,b)$ for the independent

percolation

on

periodic lattices.

Theorem 3.4. (i) If one of

$\pi_{h_{c}}(n)\approx n^{-1/\delta_{r}}$ _$or$

$\tau_{h}$

。

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holds, then both statements

as

well

as

$\mu_{h}$

。

$(\#C_{0}^{+}\geq n)\approx n^{-1/\delta}$

hold, and

$\theta(h)_{\wedge}^{\vee}L(h, \epsilon_{0})^{-1/\delta_{r}}=L(h, \epsilon_{0})^{-2/(\delta+1)},$

$\delta=2\delta_{r}-1,$

$\eta=\frac{2}{\delta_{r}}=\frac{4}{\delta+1}.$

If, in addition, for

some

$\nu>0,$

$\xi(h)\approx|h-h_{c}|^{-\nu}$ (2) holds, then

$\beta=\frac{2\nu}{\delta+1}.$

(ii)

$\bullet$ For $t\geq 2,$ $\frac{E_{h}[(\#C_{0}^{+})^{t}:\#C_{0}^{+}<\infty]}{E_{h}[(\#C_{0}^{+})^{t-1}:\#C_{0}^{+}<\infty]}\approx\xi(h)^{2}\pi_{h_{c}}(\xi(h))$,

$\bullet$ For $t>0,$ $[ \frac{1}{\chi(h)}\sum_{v\in Z^{2}}|v|^{t}\mu_{h}(Orightarrow+v, \#C_{0}^{+}<\infty)]^{1/t_{\vee}}\wedge\xi(h)$.

In addition, if (1) and (2) hold, then

$0$ For $k\geq 2,$ $\frac{E_{h}[(\#C_{0}^{+})^{k}:\#C_{0}^{+}<\infty]}{E_{h}[(\#C_{0}^{+})^{k-1}:\#C_{0}^{+}<\infty]}\approx|h-h_{c}|^{-\triangle_{k}},$

$\bullet$ For $k\geq 1,$ $[ \frac{1}{\chi(h)}\sum_{v\in Z^{2}}|v|^{k}\mu_{h}(Orightarrow+v, \#C_{0}^{+}<\infty)]^{1/k}\approx|h-h_{c}|^{-\nu_{k}},$

and

$\gamma=2\nu\frac{\delta-1}{\delta+1}, \triangle_{k}=2v\frac{\delta}{\delta+1} (k\geq 2) , v_{k}=v (k\geq 1)$

.

4 Sketch of the proof

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4.1

RSW-type estimates for crossing probabilities

In the independent percolation, the RSW lemma (Russo (1978), Seymour and Welsh (1978)$)$ gives the following estimate for crossing probabilities:

where $f_{k}(\delta)arrow 1$ as $\deltaarrow 1$

.

The following is an important consequence from the RSW

lemma. It can be obtained for Ising percolation also.

Lemma 4.1 (RSW-type estimates). For each integer $k>0$, there exists a constant $\delta_{k}$

such that for $n<L(h, \epsilon_{0})$,

(Since $\mu_{h}$ is invariant under the rotation by right angle, same estimate can be obtained

for vertical crossings.)

By the duality $\mu_{h}(A^{+}(n, n))+\mu_{h}(A^{-*}(n, n))=1$, for any $n,$

$0<\delta_{1}\leq\mu_{h}$

。$(A^{+}(n, n))\leq 1-\delta_{1}<1,$

which suggests a kind of scale invariance.

Anotherstandard tool in percolation is the Fortuin-Kasteleyn-Ginibre (FKG) inequal-ity (For the independent case, Harris (1960) already noticed the inequality.) For two configuration $\sigma,$$\sigma’\in\Omega_{V}$, we say $\sigma\leq\sigma’$ if$\sigma(v)\leq\sigma’(v)$ for all $v\in V.$

$\bullet$ An event $A\in \mathcal{F}_{V}$ is called increasing if $1_{A}(\sigma)\leq 1_{A}(\sigma’)$ whenever $\sigma\leq\sigma’$

.

For

example, $A^{+}(kn, n)$ is increasing.

$\bullet$ An event $A\in \mathcal{F}_{V}$ is called decreasing if $1_{A}(\sigma)\geq 1_{A}(\sigma’)$ whenever $\sigma\leq\sigma’$

.

For

example, $A^{-*}(kn, n)$ is decreasing.

Lemma 4.2 (the

FKG

inequality). If$A$ and $B$

are

both incresing [or both decreasing],

then $\mu_{h}(A\cap B)\geq\mu_{h}(A)\mu_{h}(B)$.

As an application, we derive a power law estimate ofthe 1-arm probability $\pi_{h_{c}}(n)$ at

the criticalpoint (especially it does not decay exponentially).

Proposition 4.3. There aile positive constants $C_{1},$ $C_{2},$ $\alpha$ such that for all _$n,$

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Proof.

First we give the upper bound. (The idea is related to that of Harris (1960).) For$j\geq 1$,

we

put $A_{j}$ $:\simeq S(4^{j+1})\backslash S(2\cdot\dot{\Phi})$, and

$X_{j}:=\{\begin{array}{l}1 If there exists a(-*)- circuit surrounding O in A_{j},0 otherwise.\end{array}$

By the mixing property and the RSW-type estimate, we can find an integer $j^{*}$ and

a

positive number $\delta$ such that for _{$j\geq j^{*},$}

$\mu_{h_{c}}(X_{j}=1|X_{1}, \ldots, X_{j-1})\geq\delta.$

$\pi_{h_{c}}(n)\leq\mu_{h_{c}}(\bigcap_{j=j^{*}}^{\lfloor\log_{4}n-1\rfloor}\{X_{j}=0\})\leq(1-\delta)^{\lfloor\log_{4}n-1\rfloor-j^{*}+1}$

Now

we

turn to the lower bound. By the RSW-type estimate,

we

have $\mu_{h_{c}}(A^{+}(n, n))\geq\delta_{1}>0.$

On $A^{+}(n, n)$,

we

look at the lowest $(+)$-crossing $L_{n}$ in $S(n)$, and put

$H(L_{n}) := \max\{y\in[-n, n] : (0, y)\in L_{n}\}.$

Then we have

$\mu_{h_{c}}(A^{+}(n, n))=\sum_{y\in[-nn]},\mu_{h}$。$(H(L_{n})=y)$

$\leq\sum_{y\in[-n,n]}\mu_{h_{c}}((0, y)rightarrow+\partial((0, y)+S(n)))$

$=(2n+1)\pi_{h_{c}}(n)$.

口

Remark 4.4. Two disjoint $(+)$-paths andone $(-*)-$

path start from neighbors of $(0, H(L_{n}))$

.

In the

inde-pendent percolation, we can obtain a better bound by the van den Berg-Kesten-Reimer inequality (see Kesten $(1987b))$. The trouble is that the inequality is not available for Ising percolation.

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Proposition 4.5. $\pi_{h_{。}}(2n)_{\wedge}^{\vee}\pi_{h_{c}}(n)$

.

Proof.

Obviously $\pi_{h_{c}}(2n)\leq\pi_{h_{c}}(n)$

.

Onthe other hand, by the RSW-type estimate and

the FKG inequality, we have

口

The following is the Ising version of Lemma in Kesten (1987a). Proposition 4.6. $\tau_{h_{c}}(n)-\vee\pi_{h_{c}}(n)^{2}.$

Proof.

For the upper bound, noting that

$\leq\mu_{h_{c}}((0,0)rightarrow\partial S(n/4)+, (n, 0)rightarrow+\partial((n, 0)+S(n/4)))$ ,

it follows from the translation-invariance and the mixing property that

$\leq\pi_{h_{c}}(n/4)^{2}+C(n/4)^{2}\cdot(n/2)\cdot e^{-\alpha n/2}.$

For the lower bound, using the RSW-type estimate and the FKG inequality,

口

Ifwe

assume

that the critical exponents $\eta$ and $\delta_{r}$ exist, then a scaling relation $\eta=\frac{2}{\delta_{r}}$

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4.2 Ising

version

of

Russo’s formula

Theorem 3.1 relates the on-critical regime to the off-critical regime, and finite regions to the whole plane. To prove it, we estimate the derivative of $\pi_{h}(n)$

.

Let $H=h/(\mathfrak{K}T)$_,

and $\mu_{H}^{N}$ denote the finite volume Gibbs distribution

on

$S(N)$ with periodic boundary

condition. For $n<N$ and $A\in \mathcal{F}_{S(n)}$, we have

$\frac{d}{dH}\mu_{H}^{N}(A)=\sum_{x\in S(N)}Cov_{\mu_{H}^{N}}(\sigma(x), 1_{A}(\sigma))$

$= \sum_{x\in S(N)}E_{\mu_{H}^{N}}[\{\sigma(x)-E_{\mu_{H}^{N}}[\sigma(x)]\}:A].$

A site $x$ is pivotalfor the event $A$ in the configuration $\sigma$ if $1_{A}(\sigma^{x})\neq 1_{A}(\sigma)$, where $\sigma^{x}$

is obtained from $\sigma$ by flipping the spin at $x$

.

Let

$Piv_{x}A:=$

{

$\sigma\in\Omega_{S(n)}:x$ is pivotal for $A$ in $\sigma$

}.

For example, $/^{\prime^{\prime^{arrow-\sim}\backslash }\backslash }$ $Piv_{x}\{Orightarrow+\partial S(n)\}=$ $\gamma,\prime/\dot{o})\wedge^{\backslash }$ $x.$ $l^{\prime^{\prime)}}\backslash _{\sim--arrow\prime}$ ’ $S(n)$

Note that $Piv_{x}A\in \mathcal{F}_{S(n)\backslash \{x\}}.$

We

assume

that $A$ is an increasing event. Note that

$A=(A\cap Piv_{x}A)\cup(A\cap(Piv_{x}A)^{c})$

In the independent percolation, we have

$\frac{d}{dp}P_{p}(A)=\sum_{x\in S(n)}P_{p}(Piv_{x}A)=E_{P_{p}}[\#$(pivotal sites for $A$)$],$

which is called Russo’s

_formula

(Russo(1981)). In the Ising percolation, we can obtain

$\frac{d}{dH}\mu_{H}^{N}(A)\geq c\sum_{x\in S(n)}\mu_{H}^{N}(Piv_{x}$み$)$,

since

$E_{\mu_{H}^{N}}[\{\sigma(x)-E_{\mu_{H}^{N}}[\sigma(x)]\}:A\cap(Piv_{x}A)^{c}]\geq 0$

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We show the strategy of the proof of Theorem 3.1(i) for the independent percolation (Kesten $(1987b)$_).

$(\subset_{J}^{\prime’}\prime\backslash \backslash /1/^{\prime\dot{o}_{1}}\sim\vee^{\backslash }\vee/^{\prime\Gamma\grave{t}_{\{}}\prime^{-arrow\sim}\backslash \iota_{\overline{S(n})}\backslash \prime/ x)$

$= \sum_{x\in S(n)}P_{p}(\backslash /^{\prime^{\prime^{\prime_{o_{1}}’}}}\vee^{\prime\backslash }/^{\nearrow^{\prime^{-\sim}}\backslash }\sim_{----\wedge^{/}}/_{S(n)}^{\backslash }R(x)\bullet,$

’

If$n<L(h, \epsilon_{0})$, then both the $(+)$-crossing probability and the $(-*)$-crossing probability

are bounded away from $0$

as

in the critical case; in a similar

manner

as

in Proposition

4.5, wehave

$P_{p}$ $(l,/\backslash _{---}乙_{}1_{\dot{O},}^{\wedge^{-\sim}}^{\backslash ,}/^{/^{/^{/}}\prime}l\prime^{/^{\prime\backslash }\backslash _{R(x_{1})}}\overline{S(}n)$

The key idea in Kesten (1987b) is to extend $(+)$-paths and $(-*)$_{-paths simultaneously:}

Roughly,

By independence,

$\frac{d}{dp}\log\pi_{p}(n)\leq C"\sum_{x\in S(n)}P_{p}(Piv_{x}A^{+}(n, n))=C"\frac{d}{dp}P_{p}(A^{+}(n, n))$

.

Integrating it from$p_{c}$ to $p(\neq p_{c})$, we have

$| \log\frac{\pi_{p}(n)}{\pi_{Pc}(n)}|\leq C"|P_{p}(A^{+}(n, n))-P_{p_{c}}(A^{+}(n, n))|.$

In the Ising case, when $A$ is the 1-arm, 4-arm, or crossing events, we can prove

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This is sufficient for

our

purpose. The key idea of the proof for the 1-arm event is to reduce

to $L_{----i_{\partial S(\mathfrak{n})}^{1}}^{---}o2_{1}^{!}$ or

by extension arguments.

4.3 Connection

lemma

Given

a

horizontal crossing $\gamma$ of $S(n)$,

we

can divide $S(n)$ into two regions; the upper

[resp. lower] one is denoted by $S^{+}(n, \gamma)$ [resp. $S^{-}(n, \gamma)$]. On $A^{+}(n, n)$, let $L_{n}$ be the

lowest $(+)$-crossing in $S(n)$. Note that $\{L_{n}=\gamma\}\in \mathcal{F}_{\gamma\cup S^{-}(n,\gamma)}$

.

In the independent case, $L_{n}$ plays the

same

role

as

the stopping time. This property together with the RSW

lemma gives

a

lower bound of the probability that there exists $a(-*)$-path from the top side of $S(n)$ to

some

point above $L_{n}$

.

It is important, for example, to estimate the

number of pivotalpoints for $A^{+}(n, n)$

.

In the Ising case, we can ‘approximately’ usethis

property, summarized

as

inthe following lemma.

Lemma 4.7 (Connection lemma). Let $V(n)=[0, n]\cross[0, kn]$

.

By a

horizontal crossing $\gamma$ of $V(n)$, we

can

divide $V(n)$ into two regions;

the upper [resp. lower] one is denoted by $V^{+}(n, \gamma)$ [resp. $V^{-}(n, \gamma)$].

Let $\gamma_{1}$ be a horizontal crossing of $[0, n]\cross[0, n]$, and $\gamma_{2}$ be

a

horizontal

crossing of $[0, n]\cross[(k-1)n, kn]$

.

There exists an integer $n_{0}$ such that if$L(h, \epsilon_{0})\geq n\geq n_{0}$, then for any $k$ and

$E\in \mathcal{F}_{V(n)}{}_{c}F\in \mathcal{F}_{\gamma_{1}\cup V^{-}(n,\gamma_{1})\cup\gamma_{2}\cup V^{+}(n,\gamma_{2})},$

we

have

$\mu_{T,h} ((\gamma_{1}+(0,1))rightarrow^{S}(\gamma_{2}+(0,-1))inV^{+}(n,\gamma_{1})\cap V^{-}(n,\gamma_{2}) E\cap F)\geq\delta_{8k}/4,$

where $s\in\{+, -*\}.$

4.4

Arm

events

We have already introduced$\pi_{h}(n)$ ($1$-armprobability) and$Piv_{0}A^{+}(n, n)$ ($4$-armevent).

More generally,

arm

events refer that there exist

some

number of crossings (arms”) of

$S(N)\backslash S(n)(N>n)$

.

For an integer $k\geq 1$ and a sequence $\sigma=(\sigma_{1}, \ldots, \sigma_{k})\in\{+, -\}^{k},$

we define the event

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that there exist $k$ disjoint crossings in _{$S(N)\backslash S(n)$}, whose signs

are

those prescribed by$\sigma$

in counterclockwise order. We also define $k$-arm events for half-planes: Let _{$S^{+}(n, N)$} _$:=$

$\{S(N)\backslash S(n)\}\cap(\mathbb{Z}_{+}\cross \mathbb{Z})$ and

$B_{k,\sigma}(n, N)=\{\partial S^{+}(n)^{k,\sigma}rightarrow\partial S^{+}(N)$ in $S^{+}(n, N)\}.$

A conjecture in Aizenman, Duplantier, and Aharony (1999) for the independent per-colation is the following:

For $k_{+},$$k_{-}\geq 1,$ $\mu_{h_{c}}(A_{k,\sigma}(n, N))\wedge(\frac{n}{N})^{(k^{2}-1)/12}$ For $k\geq 1,$ $\mu_{h_{c}}(B_{k,\sigma}(n, N))\vee\wedge(\frac{n}{N})^{(k(k+1))/6}$

where

$k_{+}:=\#\{1\leq i\leq k:\sigma_{i}=+\}$, and $k_{-};=\#\{1\leq i\leq k:\sigma i=-\}.$

For the independent percolation on the planar triangular lattice, this conjecture is proved to be true inthe sense of $\approx$ (Smirnov and Werner (2001), and Lawler, Schramm,

and Werner (2002)$)$. For two-dimensional periodic lattices, using the RSW-type

esti-mates, the conjecture is verified for $k=2,3$ in the half-plane (essentiallydone by Zhang

(1995)$)$ and $k=5$ in the whole plane (Kesten, Sidoravicius, and Zhang (1998)). (See

also Nolin (2008) and Werner (2009).$)$

Theorem 4.8. In the Ising percolation case,

we can

prove the following:

$\bullet$

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We remark that our techniques are also applicable to the Ising percolation on the triangular lattice, and they might beuseful for studyingthe scaling limit problem, posed

by B\’alint, Camia, and Meester (2010).

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